The basic idea is as follows.
We start from the large \(N\)  limit, in which the index is reproduced as the supergravity contribution in the holographic dual description [1]}.
We include finite \(N\)  corrections as contributions of
branes wrapped around topologically-trivial three-cycles in the internal space \(S^5\) .
The analysis of single-wrapping contributions
has been already done in [2]},
and it was confirmed that finite \(N\)  corrections are correctly reproduced
up to errors which can be interpreted as the multiple-wrapping contributions.
The multiple-wrapping contributions were calculated
in [3]} for the Schur index [4]} of 4d \({\cal N}=4\)  SYM,
and the analytic result of [5]} was successfully reproduced.
In this paper we improve the calculation in [3]},
and confirm that we can also calculate the
superconformal index in a similar way.
